Vol. 116 (2009)
ACTA PHYSICA POLONICA A
No. 2
Theoretical Study of Energy Levels
and Transition Probabilities of Boron Atom
Zhang Tian-yi and Zheng Neng-wu∗
Department of Chemistry, University of Science and Technology of China, Hefei, Anhui 230026, P.R. China
(Received April 20, 2009)
Though the electrons configuration for boron atom is simple and boron atom has long been of interest
for many researchers, the theoretical studies for properties of BI are not systematic, there are only few results
reported on energy levels of high excited states of boron, and transition measurements are generally restricted to
transitions involving ground states and low excited states without considering fine structure effects, provided only
multiplet results, values for transitions between high excited states are seldom performed. In this article, by using
the scheme of the weakest bound electron potential model theory calculations for energy levels of five series are
performed and with the same method we give the transition probabilities between excited states with considering
fine structure effects. The comprehensive set of calculations attempted in this paper could be of some value to
workers in the field because of the lack of published calculations for the BI systems. The perturbations coming
from foreign perturbers are taken into account in studying the energy levels. Good agreement between our results
and the accepted values taken from NIST has been obtained. We also reported some values of energy levels and
transition probabilities not existing on the NIST data bases.
PACS numbers: 01.55.+b, 31.15.–p
1. Introduction
Atomic data play a very important role in many research fields such as astrophysics, laser controlled thermonuclear fusion, and physics analytical chemistry etc.
Once the values of energy levels are known, many
properties of atomic systems can be determined. The
abundances of elements, atomic temperature, interstellar species can be inferred from the accurate knowledge
of the oscillator strengths, so the development of simple
and effective methods for calculating the energy levels
and transition probabilities. Oscillator strengths is useful for application to various other related fields.
As a light element, BI has long been of interest and
many studies of properties of BI have been carried out.
Astrophysical scientists have focused their attention on
the identity BI in the sun and the solar abundance of BI.
As we know, accurate solar abundance determination is
based on accurate values of transition rates and oscillator strengths, uncertainties in the oscillator strengths will
contribute to the uncertainties in the abundance determination, so the accurate values of transition rates and
oscillator strengths are useful for the astrophysical scientists’ research, but because the spectrum lines of neutral
boron is blended with other lines, the determination of
the solar boron abundance is uncertain. Experimentally,
considering the factors such as the strengths of the line,
line wing and cascade effects, the experimental values of
∗
corresponding author; e-mail: [email protected]
boron seldom reach a precision of 20%, sometimes only
are estimated to be accurate about 50%.
From a theoretical point of view the boron atom is an
ideal system to be studied. The ground configuration is
1s2 2s2 2p with three electrons outside a compact k-shell
core, the small number of electrons makes the theoretical
calculations simple, and accurate results are able to be
obtained.
In recent years, attention is also concentrated on excited states that are formed by excitation of 2p electron
from ground states 1s2 2s2 2p of boron atom, because the
1s2 2s2 nl excited state is similar to a one-electron system
with [He] 2s2 core. However, the studies of atom boron
are not systematic, there are only few results reported
on energy levels of high excited states of boron, and current transition measurements are generally restricted to
transitions involving ground states and low excited states
without considering fine structure effects, provide only
multiplet results, values for transitions between high excited states are seldom performed.
Experimentally, in 1966, with the phase-shift method,
radiative lifetimes of fourteen of the uv multiplets in BI
were measured by Lawrence et al., and the accuracy of
their results was estimated to be 10–20% [1]. In 1969,
seven mean lines were determined for excited neutral
and ionic states of boron by Andersen by using the foil-excitation technique [2]. By using a hollow-cathode light
source to produce the uv spectrum of BI, Goorvitch et al.
measured the transitions 2s2 2p 2P –2s2 nd 2D (n = 3–7),
2s2 2p 2P –2s2 ns 2S (n = 3–5), and 2s2 2p 2P –2s2p2 2S in
1972 [3]. Roig et al. studied the absorption spectrum of
(141)
142
Zhang Tian-yi, Zheng Neng-wu
BI by flash pyrolysis technique and observed transitions
from both the 2s2 2p 2P ground state and 2s2p2 4P state
in 1976 [4]. In 1979, by using the projective electron spectroscopy method, the ejected-electron spectra of highly
excited autoionizing levels of BI have been studied by
Rødbro et al. [5]. In 1987, for the first time the laser
spectroscopic techniques have been used to investigate
the neutral boron by Bergström et al. and the radiative
lifetimes for the 3p 2P and 4p 2P states have been reported in their work [6]. In the course of investigating
the ionization spectroscopy of boron-containing radicals,
new np (n = 5–9) Rydberg states were discovered by
Irikura et al. in 1992 [7]. In the same year, Lynam et al.
obtained the spectra of boron in a laser produced plasma
experiment and predicted transition energies, oscillator
strengths and intensities for the 1s2 2s2 2p 2P –1s2s2 2pnp,
n ≥ 2 of BI [8]. In 2001, Glab et al. reported 11 B
2s2 3s–2s2 np (n = 30–69) transition energies [9]. In the
same year, radiative lifetimes in the s and d sequences of
neutral boron were investigated both experimentally and
theoretically by Lundberg et al., they first measured the
radiative lifetimes in BI 2s2 ns 2S (n ≤ 7) and 2s2 nd 2D
(n ≤ 6) employing selective laser and then using multiconfiguration Hartree–Fock (MCHF) method to make
theoretical calculations [10].
There are also many theoretical methods developed to
obtain the energy levels and transition probabilities of
atoms and ions. Among all the methods, MCHF method
or MCHF combined with other method are the ones that
are most widely used in calculating energy levels and
transition probabilities of BI. In 1978, Dankwort and
Trefftz used a MCHF ansatz with all Breit–Pauli (BP)
corrections to calculate transition probabilities of very
highly ionized boron-like series up to Fe21+ [11]. After two years, the relativistic multiconfigurational Dirac–
Hartree–Fock (MCDHF) method were used to study the
oscillator strengths and transition energies for E1 and M1
transitions in boron by Samii et al. [12]. In 1994, with
the MCHF method, calculations have been performed for
1s2 2s2 ns 2S n = 3–6, 1s2 2s2 np 2P n = 2–6, 1s2 2s2 nd 2D
n = 3–5 states by Carlsson et al.; in this work, they used
increasing active set to calculate the transition matrix,
the largest one is 7s6p5d4f 3g [13]. In 1996, Jönsson et al.
used MCHF approach combined with configuration interaction (CI) approach to study the transition probabilities
for allowed 2s2 2pn –2s2pn+1 transitions in BI [14], after 4
years the same group used the BP Hamiltonian to calculate the configuration interaction in MCHF method, and
applications are presented for the B-like spectrum [15],
then they extend this method to study the Be-like to Ne-like sequences [16], today’s computer power make the
complex calculation process possible. From the work
mentioned above we can see that in the MCHF method
the completeness of the set of configurations used determine the accuracy of results, and the possible configurations are so many that only some can be selected for
practice which limits the accuracy of the method, the
more accurate the more configurations are needed and
the more complex the calculation will be. There are also
many other method for BI study, such as nuclear-charge-expansion (NCE) method, CI method, superposition
of configurations (SOC) method, Stieltjes imaging (SI)
method, multiconfiguration frozen-core (MC frozen-core)
method, multichannel quantum defect theory (MQDT)
and many-body perturbation theory (MBPT) [17–23],
take MQDT for example the complexity of calculation
rises quickly as the increase of the number of channels
included in.
Since the weakest bound electron potential model
(WBEPM) theory is presented, many studies have been
performed to study the atomic properties for many
systems [24–37]. In the present work, we investigate
the energy levels, transition probabilities and oscillator
strengths for BI using WBEPM theory, the results are
compared with the experimental values and the results
from other methods.
2. Theory and method
The WBEPM theory was suggested by one of the authors [38–41]. The WBEPM theory is based on the followings: (1) the considerations of successive ionization
of free particles (atom and molecule); (2) the choice of
zero of energy in quantum mechanics; (3) the separation
of the weakest bound electron (WBE) and nonweakest
bound electrons (NWBE).
In the process of successive ionization the electrons are
pealed off one by one, the WBE is the most active electron in a given system, and also most easily excited or
ionized, and all other electrons in the system are called
NWBE. The K electrons in the K-electrons system play
sooner or later as a WBE in the ionization procedure,
by removed of the first, second . . . K-th WBE, the Kelectrons system can give rise to K stage of ionization,
each stage of successive ionization processes corresponds
to the remove of a WBE from the corresponding subsystem. There is only one WBE which will be removed during the ionization process, other electrons called NWBE
will not be removed during the ionization process, so in
terms of ionization, the WBE differs in behavior from
the NWBE of the present system, so we can separate
the WBE and the NWBE, and the problem of a many-electron system can be treated as one-electron problems
of K WBEs.
The single-electron Schrödinger equation of the WBEi
is (in this paper, all the energy terms in expressions are
in Hartree units)
·
¸
1 2
− ∇i + V (ri ) ψi = εi ψi .
(1)
2
We supposed that the WBE moves in the central potential field due to the ion-core formed nucleus and NWBEs.
Considering the effect of penetrations, polarization and
shielding, we suggest the potential function of WBEi is
d(d + 1) + 2dl
−Z 0
+
V (ri ) =
,
(2)
ri
2ri2
where Z 0 is the effective nuclear charge, l is the angular
Theoretical Study of Energy Levels and Transition Probabilities of Boron Atom
quantum number of the weakest bound electron and d is
a parameter which modifies the integral quantum number ni and angular quantum number li into nonintegral
n0i and li0 .
Substituting Eq. (2) into Eq. (1), and solving the
Schrödinger equation of the WBEi, one can obtain the
expression of energy eigenvalue of WBEi:
Z 02
ε = − 02 ,
(3)
2n
0
where n is the effective principal quantum number with
n0 = n + d.
Now let us use the WBEPM theory to study atomic
levels of the excited states of BI. The electronic configuration of BI in its ground state is 1s2 2s2 2p. The 2p electrons are excited easily. The nl electron in the excited
states 1s2 2s2 nl formed by excitation of 2p electron can be
assigned as WBE. Exciting WBE to various orbitals will
produce various series of electronic configurations, each
configuration may derive several terms and each term
may go on to produce different fine structures. So the
concept of spectrum-level-like series has been introduced
to classify the energy levels [42]. Aspectrum-level-like series is a series that is composed of energy levels with the
same spectral level symbol in a given electronic configuration series of a system. From the definition, take BI
for example, the series B 1s2 2s2 nd 2D3/2 is a spectrum-level-like series.
In a given spectrum-level-like series, energy levels depend on the principal quantum number of the weakest
bound electron only
Z 02
Z 02
T (n) ≈ Tlim − 02 = Tlim −
.
(4a)
2n
2(n + d)2
Because the WBE moving in the field of the ion-core
is somewhat analogous to the valence electron in alkali
metals, and the quantum defect theory (QDT) provides
a feasible way to study levels in high Rydberg states in
alkali metals, we employ the representation of energy in
QDT to do the transformation
Z0
Znet
=
.
(4b)
n+d
n − δn
Then we get
2
Znet
T (n) = Tlim −
.
(4c)
2(n − δn )2
For neutral atoms the net nuclear charge Znet = 1 and
δn equal to the quantum defect in QDT Martin’s [43]
expression is used to determine δn :
δn (εn ) = a1 + a2 m−2 + a3 m−4 + a4 m−6 ,
(5)
in which m = n − δ0 and δ0 is the quantum defect of the
lowest level in a given level series.
It should be noted that formula (5) is only for unperturbed series, but to our knowledge, for BI atom, the
series 1s2 2s2 ns 2S are perturbed by state 1s2 2s2p2 2S
strongly, so the perturbations must be taken into account
when calculating the energy levels. On the basis of Martin’s and Langer’s [44] works, we proposed an expression for calculating levels which are perturbed by foreign
143
levels [25]:
δn (εn ) =
4
X
i=1
ai m−2(i−1) +
P
X
j=1
bj
m−2 − εj
with
m = n − δ0 ,
(6)
(7)
2(Tlim − Tj,perturber )
.
(8)
2
Znet
Here, Tj,perturber is the energy of perturbing levels, and P
is the number of the foreign perturbing levels. We used
the first (4 + P ) experiment data to fit the ai and bi by
least-squares method, then the higher ones in a series can
be predicted with these parameters.
The transition probability of (nf , lf ) to (ni , li ) for
spontaneous emission (Ef > Ei ) is
4
2
Af i = α3 (Ef − Ei )3 |hnf lf |r|ni li i| (2Lf + 1)
3
×(2Li + 1)(2Ji + 1)l> × W 2 (li Li lf Lf ; Lc 1)
εj =
×W 2 (Li Ji Lf Jf ; S1)
(9)
in a.u. [45] where l> = max(lf , li ), α is the fine structure
constant, Ef and Ei (in Hartree unit) are the energies of
(nf , lf ) and (ni , li ), respectively, Lc is the total orbital
angular momentum of atomic core and W (abcd; ef ) is the
Racah coefficient [46]. ­
¯ ¯
®
The matrix elements nf lf ¯rk ¯ ni li can be derived
as [47]:
!lf0 µ
Ã
¶l0
0
¯ k¯
­
®
2Z
2Zi0 i
f
nf +ni +lf +li
¯
¯
nf lf r ni li = (−1)
n0f
n0i
³
´


−1/2
!−lf0 −li0 −k−3
Ã
04
nf Γ n0f + lf0 + 1
Zf0
Zi0
 03

×
+ 0
n0f
ni
4Zf (nf − lf − 1)!
#−1/2 nf −lf −1 n −l −1
04
iX
i
X
ni Γ (n0i + li0 + 1)
×
03
4Zi (ni − li − 1)!
m1 =0
m2 =0

Ã
!m1 +m2 Ã
!−m1 −m2
m2
0
0
0
Zf0
Z
(−1)
Z
Z
f

− 0i
− 0i
m1 !m2 ! n0f
ni
n0f
ni
"
¡
¢
×Γ lf0 + li0 + m1 + m2 + k + 3
!
li0 − lf0 + k + m2 + 1
×
n0f − lf0 − 1 − m1 − m3
m3 =0
!
Ã
lf0 − li0 + k + m1 + 1
×
n0i − li0 − 1 − m2 − m3
Ã
!#
li0 + lf0 + k + m1 + m2 + m3 + 2
×
,
m3
s
X
Ã
(10)
where S = min (nf − lf − 1 − m1 , ni − li − 1 − m2 ) and
k > −lf0 − li0 − 3.
If we let k = 1 and i = f , the following equation can be
derived from Eq. (10), we can get the radial expectation
144
Zhang Tian-yi, Zheng Neng-wu
value of the WBE
02
3n − l0 (l0 + 1)
hrinl =
.
(11)
2Z 0
Equations (4a) and (11) constitute a set of coupled
equations
(
Z 02
T (n) ≈ Tlim − 2n
02 ,
02
(12)
3n −l0 (l0 +1)
hri =
.
2Z 0
Tlim and T (n) can be taken from the experimental
data; in this paper they are taken from NIST website [48]. The hri value can be calculated from many
theoretical methods such as Roothanna Hartree–Fock
(RHF), Hartree–Kohn–Sham (HKS), multiconfiguration
Hartree–Fock (MCHF), self-interaction-corrected local
spindensity (SIC-LSD), time-dependent Hartree–Fock
(TDHF), Hartree–Slater and numerical Coulomb approximation (NCA) etc. [49–55]. In this paper NCA is employed to evaluate hri. NCA is a good approximation
for excited states, its producing is simple and its results
agree well with other theoretical methods.
After obtaining the values of Z 0 , n0 and l0 , the matrix
element in Eq. (9) can be calculated, transition probabilities between two levels (nf , lf ) and (ni , li ) can be
calculated further.
3. Results and discussion
Energy levels, transition probabilities and oscillator
strengths of atomic BI are studied, some of the results
are listed here.
The energy levels for five spectrum-level-like series 1s2 2s2 ns 2S1/2 , 1s2 2s2 nd 2D3/2 , 1s2 2s2 nd 2D5/2 ,
1s2 2s2 nf 2F5/2 , and 1s2 2s2 nf 2F7/2 are listed in
Tables I–V, and for space reason when n > 30 some
results are omitted. Experimental data are listed in
Tables I–V for comparison. In Table I and Table II, we
list the results calculated by using multi-channel quantum defect theory (MQDT) by Liang nad Wang [22]; in
their work, the energy levels for 1s2 2s2 ns 2S1/2 (n =
2–25) and 1s2 2s2 nd 2D3/2 (n = 2–25) series are given;
from their work we can see that in MQDT the complexion
of calculation rises quickly as the increase of the number
of channels included in, relatively our calculation procedures are quite simple.
If a foreign level has the same parity and the same
quantum number J with a spectrum-level-like series, this
foreign level will perturb some energy levels in this series,
the perturbing strength is dependent on the energy difference between the perturbing level and the perturbed
level. In the treatment of the series 1s2 2s2 ns 2S1/2 , we
take into account the effects of perturbation come from
the 1s2 2s2p2 2S1/2 level. In Table I we give the results
with the consideration of foreign levels of 1s2 2s2 ns 2S1/2
series, in order to take a comparison we also give the results without the perturbations. Form Table I we can see
that the results without the perturbations are bad, the
maximal deviation is −358.4719 (n = 7); when we introduce the perturbations to our calculation, the results
TABLE I
Calculated results of energy levels (cm−1 ) for BI
1s2 2s2 ns 2S1/2 (limit is 66928.10 cm−1 ) series compared with experimental data (cm−1 ) and other results (cm−1 ).
Without perturbation
With perturbations
n
Texp [48]
a
Tcal
b
Tcal
Tother [22]
3
40039.65
40039.65
40039.65
40040.76
4
55010.181
55010.18
55010.18
55011.50
5
60146.45
60146.45
60146.45
60145.64
6
62482.23
62482.23
62482.23
62482.47
7
64156.00
63797.53
64156.00
64155.59
8
64792.07
64613.03
64767.58
64791.79
9
65270.16
65151.20
65258.17
65270.12
10
65609.35
65523.57
65602.67
65609.48
11
65791.18
65851.41
65855.68
12
65989.59
66036.42
66039.28
13
66140.58
66177.63
66179.63
14
66258.05
66287.82
66289.27
15
66351.19
66375.43
66376.50
16
66426.24
66446.22
66447.03
17
66487.60
66504.24
66504.86
18
66538.38
66552.38
66552.86
19
66580.88
66592.76
66593.14
20
66616.80
66626.97
66627.27
21
66647.43
66656.20
66656.43
22
66673.76
66681.37
66681.55
23
66696.56
66703.20
66703.34
24
66716.42
66722.26
25
66733.84
66738.99
26
66749.20
66753.77
27
66762.80
66766.88
28
66774.91
66778.56
29
66785.74
66789.02
30
66795.46
66798.41
35
66831.82
66833.67
40
66855.05
66856.28
45
66870.79
66871.65
50
66881.94
66882.56
55
66890.13
66890.59
60
66896.31
66896.67
65
66901.10
66901.38
70
66904.89
66905.11
75
66907.93
66908.11
80
66910.41
66910.56
85
66912.46
66912.58
66739.07
a
Tcal
are energy levels calculated not including perturbing level.
b
Tcal
are energy levels calculated including perturbing level. The
perturbing level is 1s2 2s2p2 2S1/2 (63560.64 cm−1 ) which is selected from experimental values in [48].
are greatly improved, the maximal deviation is −24.4923
(n = 8). From a comparison of these results, we suggest
that foreign levels should be taken into account. The
Martin expression is not available for perturbed levels, so
we derived the expression (6) to calculate the perturbed
series.
The parameters a and b for each series are listed in
Table VI, so one can easily obtain any energy levels in
the series. These parameters are obtained by fitting the
Theoretical Study of Energy Levels and Transition Probabilities of Boron Atom
TABLE II
Calculated results of energy levels
(cm−1 ) for BI 1s2 2s2 nd 2D3/2 (limit
is 66928.10 cm−1 ) series compared with
experimental data (cm−1 ) and other
results (cm−1 ).
145
TABLE III
Calculated results of energy levels
(cm−1 ) for BI 1s2 2s2 nd 2D5/2 (limit
is 66928.10 cm−1 ) series compared with
experimental data (cm−1 ).
n
Texp [48]
Tcal
n
Texp [48]
Tcal
n
Texp [48]
Tcal
Tother [22]
3
54767.80
54767.80
28
66787.66
66787.71
3
54767.63
54767.63
54767.67
4
59993.51
59993.51
29
66797.25
66797.24
4
59993.41
59993.41
59993.39
5
62485.58
62485.58
30
66805.63
66805.83
5
62485.42
62485.42
62485.45
6
63845.29
63845.29
31
66813.66
66813.60
6
63845.29
63845.29
63845.16
7
64665.34
64665.40
32
66821.00
66820.66
7
64665.30
64665.48
64665.33
8
65197.29
65197.34
33
66827.01
66827.08
8
65197.31
65197.44
65197.30
9
65561.64
65561.71
34
66832.68
66832.94
9
65561.64
65561.81
65561.69
10
65822.12
65822.12
35
66838.33
66838.30
10
65822.12
65822.21
65822.09
11
66014.58
66014.63
36
66843.28
66843.23
11
66014.58
66014.70
66014.59
12
66160.92
66160.94
37
66847.81
66847.76
12
66160.92
66161.01
66160.90
13
66274.65
66274.73
38
66851.99
66851.94
13
66274.65
66274.79
66274.68
14
66364.90
66364.97
39
66855.78
66855.80
14
66364.90
66365.01
66364.91
15
66437.62
66437.72
40
66859.37
15
66437.62
66437.76
66437.66
16
66497.15
66497.24
45
66873.81
16
66497.15
66497.28
66497.17
17
66546.47
66546.55
50
66884.13
17
66546.47
66546.58
66546.47
18
66587.77
66587.85
55
66891.77
18
66587.77
66587.88
66587.77
19
66622.72
66622.80
60
66897.57
19
66622.72
66622.82
66622.71
20
66652.52
66652.62
65
66902.09
20
66652.52
66652.64
66652.54
21
66678.17
66678.28
70
66905.68
21
66678.17
66678.30
66678.19
22
66700.43
66700.52
75
66908.57
22
66700.43
66700.53
66700.43
23
66719.79
66719.91
80
66910.94
23
66719.79
66719.92
24
66736.82
66736.92
85
66912.90
24
66736.82
66736.94
66736.83
25
66752.02
66751.94
90
66914.54
25
66752.02
66751.95
66751.84
26
66765.22
66765.25
95
66915.93
26
66765.22
66765.26
27
66777.12
66777.10
100
66917.12
27
66777.12
66777.11
28
66787.66
66787.72
29
66797.25
66797.25
30
66805.63
66805.84
31
66813.66
66813.61
32
66821.00
66820.66
33
66827.01
66827.08
34
66832.68
66832.94
35
66838.33
66838.31
36
66843.28
66843.23
37
66847.81
66847.76
38
66851.99
66851.94
39
66855.78
66855.80
40
66859.37
45
66873.81
50
66884.13
55
66891.77
60
66897.58
65
66902.09
70
66905.68
75
66908.57
80
66910.94
85
66912.90
90
66914.54
95
66915.93
100
66917.12
first (4 + P ) experiment data, by least-squares method
in expressions (5) and (6), P is the number of levels perturbing, so the choice of the values used to determine
the parameters is very important. In this work we chose
the experimental values from NIST database [48]. NIST
scientists have tried their best to collect accepted data
from the original sources, the values collected by NIST
database are the best available data at present.
A coupled Eq. (12) is derived for transition calculations. If the energy levels T (n) and the expectation values hri are known, the parameters Z 0 , n0 and l0 needed
for transition calculations can be obtained. The values
of T (n) are also taken from NIST database [48] and hri
evaluated using NCA method.
The spin-allowed transition probabilities and oscillator strength lines in BI are calculated, and the results
are listed in Table VII, we compared our results to
NIST data [48], the accuracy rating of NIST values are
given in the sixth column in Table VII. Presently the
study for transitions between individual lines for BI are
quite limited both experimentally and theoretically, and
the existing values are generally for transitions involving ground states and lower excited states, transitions
involving highly excited states are not studied by other
methods.
146
Zhang Tian-yi, Zheng Neng-wu
TABLE IV
Calculated results of energy levels
(cm−1 ) for BI 1s2 2s2 nf 2F5/2 (limit is
66928.10 cm−1 ) series compared with
experimental data (cm−1 ).
TABLE V
Calculated results of energy levels
(cm−1 ) for BI 1s2 2s2 nf 2F7/2 (limit is
66928.10 cm−1 ) series compared with
experimental data (cm−1 ).
n
Texp [48]
Tcal
n
Tcal
n
Texp [48]
Tcal
n
4
60031.03
60031.03
25
66752.30
4
60031.03
60031.03
25
66752.30
5
62516.52
62516.52
26
66765.57
5
62516.52
62516.52
26
66765.57
6
63866.33
63866.33
27
66777.40
6
63866.33
63866.33
27
66777.40
7
64679.74
64679.74
28
66787.97
7
64679.74
64679.74
28
66787.97
8
65207.50
65207.39
29
66797.48
8
65207.50
65207.39
29
66797.48
9
65569.06
65568.98
30
66806.04
9
65569.06
65568.98
30
66806.04
10
65827.38
65827.52
35
66838.44
10
65827.38
65827.52
35
66838.44
11
66018.75
66018.75
40
66859.46
11
66018.75
40
66859.46
12
66164.14
45
66873.87
12
66164.14
45
66873.87
13
66277.27
50
66884.18
13
66277.27
50
66884.18
14
66367.01
55
66891.80
14
66367.01
55
66891.80
15
66439.40
60
66897.60
15
66439.40
60
66897.60
16
66498.63
65
66902.11
16
66498.63
65
66902.11
17
66547.71
70
66905.69
17
66547.71
70
66905.69
18
66588.83
75
66908.58
18
66588.83
75
66908.58
19
66623.63
80
66910.95
19
66623.63
80
66910.95
20
66653.34
85
66912.91
20
66653.34
85
66912.91
21
66678.90
90
66914.55
21
66678.90
90
66914.55
22
66701.05
95
66915.94
22
66701.05
95
66915.94
23
66720.38
100
66917.12
23
66720.38
100
66917.12
24
66737.34
24
66737.34
Texp [48]
Texp [48]
Tcal
TABLE VI
Parameters a and b of every tables.
Table
Table
Table
Table
Table
Table
a
Ia
Ib
II
III
IV
V
a1
a2
a3
1.24532
0.93534
0.04127
0.04207
0.01563
0.01563
−7.61248
0.51214
−0.26264
−0.31054
−0.11462
−0.11462
61.96560
−4.16833
0.01118
0.81200
1.37123
1.37123
without perturbation;
b
a4
b1
b2 b3 b4
−144.14765
11.27022 0.00074
−11.98405
−15.91133
−10.70191
−10.70191
with perturbation
However, the transitions probabilities for individual
line have many applications in the corresponding fields,
so present calculation is very significant. Good agreement is obtained from the comparison in this paper. We
predict many values in the case of the NIST values not
existing, the good agreement enables us to believe the
predicted results are reliable.
Because most theoretical methods are only used to obtain transition probabilities for multiplets, in order to
make the comparison with the other method and test the
reliability of our method further, we also calculate transition probabilities and oscillator strengths BI for multi-
plets. The results are listed in Table VIII. In Table VIII
the transition rates performed using the MCHF method
[10] are listed in the fifth column and the oscillator
strength calculated by other method are given in the
sixth column. From the comparison our results are generally closer to the accepted values than the others and
in the case of the accepted values do not exist, our results
are close to the others.
In conclusion, the WBEPM theory is successful in
studying the energy levels and transition probabilities for
BI. The calculation procedure is simple and the results
are accurate.
Theoretical Study of Energy Levels and Transition Probabilities of Boron Atom
TABLE VII
Transition probabilities and oscillator strengths for individual lines of BI and comparison with
accepted values.
Transition
0
3s 2S1/2 –3p 2P1/2
2
2 0
3s S1/2 –3p P3/2
0
3s 2S1/2 –4p 2P1/2
2
2 0
3s S1/2 –4p P3/2
0
4s 2S1/2 –3p 2P1/2
0
4s 2S1/2 –3p 2P3/2
2
2 0
4s S1/2 –4p P1/2
0
4s 2S1/2 –4p 2P3/2
0
5s 2S1/2 –3p 2P1/2
2
2 0
5s S1/2 –3p P3/2
0
5s 2S1/2 –4p 2P1/2
0
5s 2S1/2 –4p 2P3/2
2
2 0
6s S1/2 –3p P1/2
0
6s 2S1/2 –3p 2P3/2
2
2 0
6s S1/2 –4p P1/2
0
6s 2S1/2 –4p 2P3/2
0
7s 2S1/2 –3p 2P1/2
2
2 0
7s S1/2 –3p P3/2
0
7s 2S1/2 –4p 2P1/2
0
7s 2S1/2 –4p 2P3/2
2
2 0
8s S1/2 –3p P1/2
0
8s 2S1/2 –3p 2P3/2
0
8s 2S1/2 –4p 2P1/2
2
2 0
8s S1/2 –4p P3/2
0
9s 2S1/2 –3p 2P1/2
0
9s 2S1/2 –3p 2P3/2
2
2 0
9s S1/2 –4p P1/2
0
9s 2S1/2 –4p 2P3/2
0
10s 2S1/2 –3p 2P1/2
2
2 0
10s S1/2 –3p P3/2
0
10s 2S1/2 –4p 2P1/2
0
10s 2S1/2 –4p 2P3/2
2 0
2
3p P1/2 –3d D3/2
0
3p 2P1/2
–4d 2D3/2
0
3p 2P1/2
–5d 2D3/2
2 0
3p P1/2 –6d 2D3/2
0
3p 2P1/2
–7d 2D3/2
0
3p 2P1/2
–8d 2D3/2
2 0
3p P1/2 –9d 2D3/2
0
3p 2P1/2
–10d 2D3/2
0
3p 2P3/2
–3d 2D3/2
2 0
3p P3/2 –3d 2D5/2
Present transition NIST transition Present NIST oscillator
probabilities
probabilities
oscillator
strengths
8 −1
8 −1
[10 s ]
[10 s ] [48] strengths
[48]
1.73×10−1
1.73×10−1
3.84×10−3
3.85×10−3
4.91×10−2
9.83×10−2
2.65×10−2
2.66×10−2
1.93×10−2
3.85×10−2
1.21×10−2
2.42×10−2
8.76×10−3
1.75×10−2
4.33×10−3
8.65×10−3
3.93×10−3
7.87×10−3
2.60×10−3
5.21×10−3
3.54×10−3
7.08×10−3
2.02×10−3
4.03×10−3
2.47×10−3
4.94×10−3
1.35×10−3
2.70×10−3
1.76×10−3
3.51×10−3
9.39×10−4
1.88×10−3
1.13×10−1
8.66×10−4
3.13×10−4
6.88×10−4
6.91×10−4
5.82×10−4
4.70×10−4
3.73×10−4
2.26×10−2
1.36×10−1
0.174
0.174
0.051
0.103
0.0162
0.0324
0.115
0.023
0.138
3.53×10−1
7.07×10−1
1.83×10−3
3.67×10−3
1.80×10−1
1.80×10−1
5.16×10−1
1.03
2.17×10−2
2.17×10−2
3.25×10−1
3.26×10−1
6.82×10−3
6.82×10−3
2.94×10−2
2.94×10−2
2.44×10−3
2.44×10−3
9.62×10−3
9.62×10−3
2.03×10−3
2.03×10−3
6.16×10−3
6.16×10−3
1.33×10−3
1.33×10−3
3.61×10−3
3.61×10−3
9.11×10−4
9.11×10−4
2.30×10−3
2.30×10−3
8.96×10−1
2.00×10−3
4.87×10−4
8.89×10−4
8.03×10−4
6.35×10−4
4.91×10−4
3.77×10−4
8.96×10−2
8.06×10−1
Accuraccy
of NIST
[48]
0.355
0.710
C
C
0.19
0.189
C
C
0.0183
0.0183
C
C
0.91
C
0.091
0.819
C
C
147
148
Zhang Tian-yi, Zheng Neng-wu
TABLE VII (cont.)
TABLE VII (cont.)
Present transition
Present oscillator
probabilities [108 s−1 ]
strengths
0
3p 2P3/2
–4d 2D3/2
1.72×10−4
1.99×10−4
0
3p 2P3/2
–4d 2D5/2
1.03×10−3
0
3p 2P3/2
–5d 2D3/2
6.31×10−5
0
3p 2P3/2
–5d 2D5/2
Transition
2
0
3p P3/2
–6d 2D3/2
2 0
3p P3/2 –6d 2D5/2
0
3p 2P3/2
–7d 2D3/2
2 0
3p P3/2 –7d 2D5/2
0
3p 2P3/2
–8d 2D3/2
2 0
3p P3/2 –8d 2D5/2
0
3p 2P3/2
–9d 2D3/2
0
3p 2P3/2
–9d 2D5/2
0
3p 2P3/2
–10d 2D3/2
0
3p 2P3/2
–10d 2D5/2
2 0
4p P1/2 –3d 2D3/2
0
4p 2P1/2
–4d 2D3/2
2 0
4p P1/2 –5d 2D3/2
0
4p 2P1/2
–6d 2D3/2
0
4p 2P1/2
–7d 2D3/2
0
4p 2P1/2
–8d 2D3/2
0
4p 2P1/2
–9d 2D3/2
2 0
4p P1/2 –10d 2D3/2
0
4p 2P3/2
–3d 2D3/2
2 0
4p P3/2 –3d 2D5/2
0
4p 2P3/2
–4d 2D3/2
2 0
4p P3/2 –4d 2D5/2
0
4p 2P3/2
–5d 2D3/2
0
4p 2P3/2
–5d 2D5/2
0
4p 2P3/2
–6d 2D3/2
0
4p 2P3/2
–6d 2D5/2
2 0
4p P3/2 –7d 2D3/2
0
4p 2P3/2
–7d 2D5/2
2 0
4p P3/2 –8d 2D3/2
0
4p 2P3/2
–8d 2D5/2
2 0
4p P3/2 –9d 2D3/2
0
4p 2P3/2
–9d 2D5/2
0
4p 2P3/2
–10d 2D3/2
0
4p 2P3/2
–10d 2D5/2
0
3d 2D3/2 –4f 2F5/2
2
2 0
3d D3/2 –5f F5/2
0
3d 2D3/2 –6f 2F5/2
2
2 0
3d D3/2 –7f F5/2
0
3d 2D3/2 –8f 2F5/2
2
2 0
3d D3/2 –9f F5/2
0
3d 2D3/2 –10f 2F5/2
0
3d 2D3/2 –11f 2F5/2
0
3d 2D5/2 –4f 2F5/2
0
3d 2D5/2 –4f 2F7/2
2
2 0
3d D5/2 –5f F5/2
0
3d 2D5/2 –5f 2F7/2
2
2 0
3d D5/2 –6f F5/2
0
3d 2D5/2 –6f 2F7/2
2
2 0
3d D5/2 –7f F5/2
Transition
Present transition
Present oscillator
probabilities [108 s−1 ]
strengths
0
3d 2D5/2 –7f 2F7/2
1.10×10−2
2.24×10−2
1.79×10−3
0
3d 2D5/2 –8f 2F5/2
4.56×10−4
6.28×10−4
4.91×10−5
0
3d 2D5/2 –8f 2F7/2
6.85×10−3
1.26×10−2
3.76×10−4
4.40×10−4
0
3d 2D5/2 –9f 2F5/2
3.05×10−4
3.92×10−4
1.38×10
−4
8.93×10
−5
4.57×10
−3
7.83×10−3
8.29×10
−4
8.04×10
−4
2.14×10
−4
2.62×10−4
1.39×10
−4
8.06×10
−5
3.20×10
−3
5.24×10−3
8.30×10
−4
7.25×10
−4
1.57×10
−4
1.86×10−4
1.17×10
−4
6.37×10
−5
1.08×10
−7
1.72×10−2
7.02×10
−4
5.74×10
−4
2.44×10
−2
8.61×10−1
1.23×10−2
1.85×10−1
7.07×10−3
7.24×10−2
4.46×10−3
3.69×10−2
3.01×10
−3
2.17×10−2
2.12×10
−3
1.40×10−2
1.56×10
−3
9.68×10−3
7.68×10
−9
8.18×10−4
1.15×10
−7
1.64×10−2
1.74×10−3
4.10×10−2
2.61×10−2
8.20×10−1
8.80×10−4
8.80×10−3
1.32×10
−2
1.76×10−1
5.05×10
−4
3.45×10−3
7.57×10
−3
6.89×10−2
3.18×10
−4
1.76×10−3
4.78×10
−3
3.51×10−2
2.15×10
−4
1.04×10−3
3.22×10−3
2.07×10−2
1.51×10−4
6.67×10−4
2.27×10−3
1.33×10−2
1.12×10
−4
4.61×10−4
6.37×10
−4
1.06×10−2
2.19×10
−7
5.10×10−2
6.68×10
−3
7.87×10−1
4.11×10
−3
1.92×10−1
2.64×10
−3
8.02×10−2
1.80×10−3
4.25×10−2
1.28×10−3
2.57×10−2
9.43×10−4
1.70×10−2
3.03×10
−5
7.53×10−4
6.06×10
−4
1.13×10−2
1.54×10
−8
2.42×10−3
2.31×10
−7
4.83×10−2
4.77×10
−4
3.75×10−2
7.15×10−3
7.50×10−1
2.94×10−4
9.14×10−3
4.40×10−3
1.83×10−1
1.89×10
−4
3.82×10−3
2.83×10
−3
7.64×10−2
1.28×10
−4
2.03×10−3
1.93×10
−3
4.05×10−2
9.12×10
−5
1.23×10−3
1.37×10
−3
2.45×10−2
9.43×10−5
4.92×10−5
5.66×10−4
4.43×10−4
7.48×10−5
3.78×10−5
4.49×10
−4
3.41×10
−4
1.80×10
−2
1.48×10
−1
2.04×10
−2
1.19×10
−3
9.92×10
−5
1.26
1.62×10
−2
8.10×10
−4
2.50×10−6
1.58×10−5
2.68×10−6
1.46×10−5
8.74×10−6
4.34×10−5
1.15×10
−5
5.33×10
−5
1.80×10
−3
2.97×10
−2
1.62×10
−2
1.78×10
−1
4.08×10
−3
1.26×10
−1
2.45×10
−2
2.38×10
−4
1.13
1.62×10
−3
1.43×10−3
1.46×10−2
1.97×10−5
8.04×10−5
1.18×10−4
7.23×10−4
4.80×10
−7
1.52×10
−6
2.94×10
−6
1.40×10
−5
5.52×10
−7
1.51×10
−6
3.35×10
−6
1.37×10
−5
1.77×10
−6
4.40×10
−6
1.06×10
−5
3.96×10
−5
2.32×10−6
5.39×10−6
1.39×10−5
4.85×10−5
1.26×10−1
1.02
3.98×10
−2
1.84×10
−2
1.03×10
−2
6.39×10
−3
4.27×10
−3
1.49×10
−1
5.01×10
−2
2.35×10
−2
1.32×10
−2
8.23×10
−3
2.99×10−3
5.50×10−3
2.20×10−3
3.91×10−3
9.02×10−3
4.88×10−2
1.35×10
−1
9.76×10
−1
2.84×10
−3
7.09×10
−3
4.26×10
−2
1.42×10
−1
1.32×10
−3
2.39×10
−3
1.98×10
−2
4.77×10
−2
7.34×10
−4
1.12×10
−3
2
2
0
3d D5/2 –9f F7/2
2
2 0
3d D5/2 –10f F5/2
0
3d 2D5/2 –10f 2F7/2
2
2 0
3d D5/2 –11f F5/2
0
4d 2D3/2 –4f 2F5/2
2
2 0
4d D3/2 –5f F5/2
0
4d 2D3/2 –6f 2F5/2
0
4d 2D3/2 –7f 2F5/2
0
4d 2D3/2 –8f 2F5/2
2
2 0
4d D3/2 –9f F5/2
0
4d 2D3/2 –10f 2F5/2
2
2 0
4d D3/2 –11f F5/2
0
4d 2D5/2 –4f 2F5/2
2
2 0
4d D5/2 –4f F7/2
0
4d 2D5/2 –5f 2F5/2
0
4d 2D5/2 –5f 2F7/2
0
4d 2D5/2 –6f 2F5/2
2
2 0
4d D5/2 –6f F7/2
0
4d 2D5/2 –7f 2F5/2
2
2 0
4d D5/2 –7f F7/2
0
4d 2D5/2 –8f 2F5/2
2
2 0
4d D5/2 –8f F7/2
0
4d 2D5/2 –9f 2F5/2
0
4d 2D5/2 –9f 2F7/2
0
4d 2D5/2 –10f 2F5/2
0
4d 2D5/2 –10f 2F7/2
2
2 0
4d D5/2 –11f F5/2
0
5d 2D3/2 –4f 2F5/2
2
2 0
5d D3/2 –5f F5/2
0
5d 2D3/2 –6f 2F5/2
2
2 0
5d D3/2 –7f F5/2
0
5d 2D3/2 –8f 2F5/2
0
5d 2D3/2 –9f 2F5/2
0
5d 2D3/2 –10f 2F5/2
0
5d 2D3/2 –11f 2F5/2
2
2 0
5d D5/2 –4f F5/2
0
5d 2D5/2 –4f 2F7/2
2
2 0
5d D5/2 –5f F5/2
0
5d 2D5/2 –5f 2F7/2
2
2 0
5d D5/2 –6f F5/2
0
5d 2D5/2 –6f 2F7/2
0
5d 2D5/2 –7f 2F5/2
0
5d 2D5/2 –7f 2F7/2
2
2 0
5d D5/2 –8f F5/2
0
5d 2D5/2 –8f 2F7/2
2
2 0
5d D5/2 –9f F5/2
0
5d 2D5/2 –9f 2F7/2
2
2 0
5d D5/2 –10f F5/2
0
5d 2D5/2 –10f 2F7/2
Theoretical Study of Energy Levels and Transition Probabilities of Boron Atom
TABLE VII (cont.)
TABLE VII (cont.)
Present transition
Present oscillator
probabilities [108 s−1 ]
strengths
0
5d 2D5/2 –11f 2F5/2
6.74×10−5
8.09×10−4
0
6d 2D3/2 –4f 2F5/2
2.76×10−4
1.90×10−3
4.92×10
−4
2.79×10
−2
1.65×10
−7
8.39×10
−2
2.35×10
−3
7.57×10
−1
1.61×10
−3
1.95×10
−1
Transition
2
2
0
6d D3/2 –5f F5/2
2
2 0
6d D3/2 –6f F5/2
0
6d 2D3/2 –7f 2F5/2
2
2 0
6d D3/2 –8f F5/2
0
6d 2D3/2 –9f 2F5/2
0
6d 2D3/2 –10f 2F5/2
0
6d 2D3/2 –11f 2F5/2
2
2 0
6d D5/2 –4f F5/2
0
6d 2D5/2 –4f 2F7/2
2
2 0
6d D5/2 –5f F5/2
0
6d 2D5/2 –5f 2F7/2
2
2 0
6d D5/2 –6f F5/2
0
6d 2D5/2 –6f 2F7/2
0
6d 2D5/2 –7f 2F5/2
0
6d 2D5/2 –7f 2F7/2
0
6d 2D5/2 –8f 2F5/2
2
2 0
6d D5/2 –8f F7/2
0
6d 2D5/2 –9f 2F5/2
2
2 0
6d D5/2 –9f F7/2
0
6d 2D5/2 –10f 2F5/2
2
2 0
6d D5/2 –10f F7/2
0
6d 2D5/2 –11f 2F5/2
0
7d 2D3/2 –4f 2F5/2
0
7d 2D3/2 –5f 2F5/2
0
7d 2D3/2 –6f 2F5/2
2
2 0
7d D3/2 –7f F5/2
0
7d 2D3/2 –8f 2F5/2
2
2 0
7d D3/2 –9f F5/2
0
7d 2D3/2 –10f 2F5/2
2
2 0
7d D3/2 –11f F5/2
0
7d 2D5/2 –4f 2F5/2
0
7d 2D5/2 –4f 2F7/2
0
7d 2D5/2 –5f 2F5/2
2
2 0
7d D5/2 –5f F7/2
0
7d 2D5/2 –6f 2F5/2
2
2 0
7d D5/2 –6f F7/2
0
7d 2D5/2 –7f 2F5/2
2
2 0
7d D5/2 –7f F7/2
0
7d 2D5/2 –8f 2F5/2
0
7d 2D5/2 –8f 2F7/2
0
7d 2D5/2 –9f 2F5/2
0
7d 2D5/2 –9f 2F7/2
2
2 0
7d D5/2 –10f F5/2
0
7d 2D5/2 –10f 2F7/2
2
2 0
7d D5/2 –11f F5/2
0
8d 2D3/2 –4f 2F5/2
2
2 0
8d D3/2 –5f F5/2
0
8d 2D3/2 –6f 2F5/2
0
8d 2D3/2 –7f 2F5/2
0
8d 2D3/2 –8f 2F5/2
0
8d 2D3/2 –9f 2F5/2
1.11×10−3
8.43×10−2
7.98×10−4
4.57×10−2
5.91×10−4
2.81×10−2
1.32×10
−5
1.36×10
−4
2.63×10
−4
2.03×10
−3
2.34×10
−5
1.99×10
−3
4.69×10
−4
2.99×10
−2
1.18×10
−8
4.00×10
−3
1.77×10
−7
7.99×10
−2
1.67×10−4
3.61×10−2
2.51×10−3
7.21×10−1
1.15×10−4
9.29×10−3
1.73×10
−3
1.86×10
−1
7.96×10
−5
4.02×10
−3
1.19×10
−3
8.03×10
−2
5.70×10
−5
2.18×10
−3
8.55×10
−4
4.35×10
−2
4.22×10
−5
1.34×10
−3
1.47×10−4
6.82×10−4
2.42×10−4
5.24×10−3
3.17×10−4
4.96×10−2
1.08×10
−7
1.16×10
−1
9.77×10
−4
7.47×10
−1
7.20×10
−4
1.98×10
−1
5.24×10
−4
8.73×10
−2
3.90×10
−4
4.79×10
−2
6.98×10−6
4.87×10−5
1.40×10−4
7.30×10−4
1.15×10−5
3.74×10−4
2.30×10
−4
5.61×10
−3
1.51×10
−5
3.54×10
−3
3.02×10
−4
5.31×10
−2
7.63×10
−9
5.52×10
−3
1.14×10
−7
1.10×10
−1
6.98×10
−5
3.56×10
−2
1.05×10−3
7.12×10−1
5.14×10−5
9.44×10−3
7.71×10−4
1.89×10−1
3.75×10
−5
4.16×10
−3
5.62×10
−4
8.32×10
−2
2.79×10
−5
2.28×10
−3
8.79×10−5
3.29×10−4
1.38×10−4
1.92×10−3
−4
9.60×10−3
1.99×10−4
7.41×10−2
6.80×10−8
1.47×10−1
4.62×10−4
7.51×10−1
1.70×10
149
Transition
Present transition
Present oscillator
probabilities [108 s−1 ]
strengths
0
8d 2D3/2 –10f 2F5/2
3.58×10−4
2.03×10−1
0
8d 2D3/2 –11f 2F5/2
2.70×10−4
8.99×10−2
4.19×10
−6
2.35×10−5
8.37×10
−5
3.53×10−4
6.58×10
−6
1.37×10−4
1.32×10
−4
2.06×10−3
8.10×10−6
6.86×10−4
1.62×10−4
1.03×10−2
9.47×10−6
5.30×10−3
1.89×10
−4
7.95×10−2
4.88×10
−9
7.02×10−3
7.33×10
−8
1.40×10−1
3.30×10
−5
3.58×10−2
4.95×10
−4
7.16×10−1
2.55×10
−5
9.65×10−3
3.83×10−4
1.93×10−1
1.93×10−5
4.28×10−3
5.73×10−5
1.87×10−4
8.75×10
−5
9.43×10−4
1.03×10
−4
3.59×10−3
1.14×10
−4
1.47×10−2
1.27×10
−4
1.01×10−1
4.35×10
−8
1.78×10−1
2.41×10
−4
7.67×10−1
1.92×10−4
2.07×10−1
2.73×10−6
1.34×10−5
5.46×10−5
2.01×10−4
4.17×10
−6
6.73×10−5
8.33×10
−5
1.01×10−3
4.92×10
−6
2.56×10−4
9.83×10
−5
3.85×10−3
5.45×10
−6
1.05×10−3
1.09×10−4
1.58×10−2
6.04×10−6
7.22×10−3
1.21×10−4
1.08×10−1
3.11×10
−9
8.46×10−3
4.66×10
−8
1.69×10−1
1.72×10
−5
3.65×10−2
2.58×10
−4
7.30×10−1
1.37×10
−5
9.84×10−3
3.95×10
−5
1.18×10−4
5.91×10−5
5.41×10−4
6.79×10−5
1.77×10−3
7.25×10−5
5.55×10−3
7.68×10
−5
2.03×10−2
8.26×10
−5
1.29×10−1
2
2
0
8d D5/2 –4f F5/2
2
2 0
8d D5/2 –4f F7/2
0
8d 2D5/2 –5f 2F5/2
2
2 0
8d D5/2 –5f F7/2
0
8d 2D5/2 –6f 2F5/2
0
8d 2D5/2 –6f 2F7/2
0
8d 2D5/2 –7f 2F5/2
2
2 0
8d D5/2 –7f F7/2
0
8d 2D5/2 –8f 2F5/2
2
2 0
8d D5/2 –8f F7/2
0
8d 2D5/2 –9f 2F5/2
2
2 0
8d D5/2 –9f F7/2
0
8d 2D5/2 –10f 2F5/2
0
8d 2D5/2 –10f 2F7/2
0
8d 2D5/2 –11f 2F5/2
0
9d 2D3/2 –4f 2F5/2
0
9d 2D3/2 –5f 2F5/2
2
2 0
9d D3/2 –6f F5/2
0
9d 2D3/2 –7f 2F5/2
2
2 0
9d D3/2 –8f F5/2
0
9d 2D3/2 –9f 2F5/2
2
2 0
9d D3/2 –10f F5/2
0
9d 2D3/2 –11f 2F5/2
0
9d 2D5/2 –4f 2F5/2
0
9d 2D5/2 –4f 2F7/2
2
2 0
9d D5/2 –5f F5/2
0
9d 2D5/2 –5f 2F7/2
2
2 0
9d D5/2 –6f F5/2
0
9d 2D5/2 –6f 2F7/2
2
2 0
9d D5/2 –7f F5/2
0
9d 2D5/2 –7f 2F7/2
0
9d 2D5/2 –8f 2F5/2
0
9d 2D5/2 –8f 2F7/2
2
2 0
9d D5/2 –9f F5/2
0
9d 2D5/2 –9f 2F7/2
2
2 0
9d D5/2 –10f F5/2
0
9d 2D5/2 –10f 2F7/2
2
2 0
9d D5/2 –11f F5/2
0
10d 2D3/2 –4f 2F5/2
0
10d 2D3/2 –5f 2F5/2
0
10d 2D3/2 –6f 2F5/2
0
10d 2D3/2 –7f 2F5/2
2
2 0
10d D3/2 –8f F5/2
0
10d 2D3/2 –9f 2F5/2
C — estimated accuracy ≤ 25%
150
Zhang Tian-yi, Zheng Neng-wu
TABLE VIII
Transition probabilities and oscillator strengths for multiplets of BI and comparison with accepted values and
other results.
Transition
Present transition NIST transition
NIST oscillator
Present oscillator
probabilities
probabilities
strength
strength
8 −1
8 −1
[10 s ]
[48] [10 s ]
[48]
3s 2S–3p 2P 0
1.73×10−1
3s 2S–4p 2P 0
4s 2S–3p 2P 0
3.85×10−3
1.47×10−1
4s 2S–4p 2P 0
5s 2S–3p 2P 0
2.66×10−2
5.77×10−2
5s 2S–4p 2P 0
6s 2S–3p 2P 0
6s 2S–4p 2P 0
7s 2S–3p 2P 0
7s 2S–4p 2P 0
8s 2S–3p 2P 0
8s 2S–4p 2P 0
9s 2S–3p 2P 0
9s 2S–4p 2P 0
10s 2S–3p 2P 0
10s 2S–4p 2P 0
3p 2P 0 –3d 2D
3.63×10−2
2.63×10−2
1.30×10−2
1.18×10−2
7.81×10−3
1.06×10−2
6.05×10−3
7.41×10−3
4.05×10−3
5.27×10−3
2.82×10−3
1.36×10−1
3p 2P 0 –4d 2D
3p 2P 0 –5d 2D
3p 2P 0 –6d 2D
3p 2P 0 –7d 2D
3p 2P 0 –8d 2D
3p 2P 0 –9d 2D
3p 2P 0 –10d 2D
4p 2P 0 –3d 2D
4p 2P 0 –4d 2D
4p 2P 0 –5d 2D
4p 2P 0 –6d 2D
4p 2P 0 –7d 2D
4p 2P 0 –8d 2D
4p 2P 0 –9d 2D
4p 2P 0 –10d 2D
3d 2D–4f 2F 0
3d 2D–5f 2F 0
3d 2D–6f 2F 0
3d 2D–7f 2F 0
3d 2D–8f 2F 0
1.04×10−3
3.76×10−4
8.28×10−4
8.30×10−4
7.01×10−4
5.66×10−4
4.48×10−4
1.80×10−2
2.45×10−2
1.43×10−3
1.18×10−4
2.95×10−6
3.30×10−6
1.06×10−5
1.39×10−5
1.35×10−1
4.26×10−2
1.98×10−2
1.10×10−2
6.85×10−3
0.174
1.06
Other
transition
probabilities
Other
oscillator
strength
1.06
0.1720 [10]
1.126 [56]
1.060 [21]
1.199 [57]
0.18
0.002066 [10]
0.1657 [10]
0.0182
0.02603 [10]
0.05229 [10]
5.49×10−3
0.15
0.0486
1.80×10−1
1.55
0.138
2.17×10−2
3.26×10−1
6.82×10−3
2.94×10−2
2.44×10−3
9.62×10−3
2.03×10−3
6.16×10−3
1.33×10−3
3.61×10−3
9.11×10−4
2.30×10−3
8.96×10−1
2.00×10−3
4.88×10−4
8.92×10−4
8.05×10−4
6.36×10−4
4.92×10−4
3.78×10−4
1.78×10−1
1.26
1.62×10−2
8.06×10−4
1.56×10−5
1.50×10−5
4.38×10−5
5.37×10−5
1.02
1.49×10−1
5.01×10−2
2.35×10−2
0.202 [56]
0.187 [21]
0.188 [57]
0.0198 [56]
0.0187 [21]
0.0182 [57]
0.03745 [10]
0.01817 [10]
0.008948 [10]
0.01895 [10]
0.910
0.1278 [10]
0.0001323 [10]
0.002324 [10]
0.002686 [10]
0.01673 [10]
0.02525 [10]
0.000916 [10]
0.00002152 [10]
0.1209 [10]
0.03706 [10]
0.844 [56]
0.889 [21]
0.786 [57]
Theoretical Study of Energy Levels and Transition Probabilities of Boron Atom
151
TABLE VIII (cont.)
Transition
3d 2D–9f 2F 0
3d 2D–10f 2F 0
3d 2D–11f 2F 0
4d 2D–4f 2F 0
4d 2D–5f 2F 0
4d 2D–6f 2F 0
4d 2D–7f 2F 0
4d 2D–8f 2F 0
4d 2D–9f 2F 0
4d 2D–10f 2F 0
4d 2D–11f 2F 0
5d 2D–4f 2F 0
5d 2D–5f 2F 0
5d 2D–6f 2F 0
5d 2D–7f 2F 0
5d 2D–8f 2F 0
5d 2D–9f 2F 0
5d 2D–10f 2F 0
5d 2D–11f 2F 0
6d 2D–4f 2F 0
6d 2D–5f 2F 0
6d 2D–6f 2F 0
6d 2D–7f 2F 0
6d 2D–8f 2F 0
6d 2D–9f 2F 0
6d 2D–10f 2F 0
6d 2D–11f 2F 0
7d 2D–4f 2F 0
7d 2D–5f 2F 0
7d 2D–6f 2F 0
7d 2D–7f 2F 0
7d 2D–8f 2F 0
7d 2D–9f 2F 0
7d 2D–10f 2F 0
7d 2D–11f 2F 0
8d 2D–4f 2F 0
8d 2D–5f 2F 0
8d 2D–6f 2F 0
8d 2D–7f 2F 0
8d 2D–8f 2F 0
8d 2D–9f 2F 0
8d 2D–10f 2F 0
8d 2D–11f 2F 0
9d 2D–4f 2F 0
9d 2D–5f 2F 0
9d 2D–6f 2F 0
9d 2D–7f 2F 0
Present transition NIST transition
NIST oscillator
Present oscillator
probabilities
probabilities
strength
strength
[108 s−1 ]
[48] [108 s−1 ]
[48]
4.57×10−3
3.20×10−3
2.36×10−3
1.16×10−7
2.61×10−2
1.32×10−2
7.57×10−3
4.78×10−3
3.22×10−3
2.27×10−3
1.67×10−3
6.36×10−4
2.33×10−7
7.15×10−3
4.40×10−3
2.83×10−3
1.93×10−3
1.37×10−3
1.01×10−3
2.76×10−4
4.92×10−4
1.77×10−7
2.51×10−3
1.73×10−3
1.19×10−3
8.55×10−4
6.33×10−4
1.47×10−4
2.42×10−4
3.17×10−4
1.15×10−7
1.05×10−3
7.71×10−4
5.62×10−4
4.18×10−4
8.79×10−5
1.38×10−4
1.70×10−4
1.99×10−4
7.30×10−8
4.95×10−4
3.83×10−4
2.89×10−4
5.73×10−5
8.75×10−5
1.03×10−4
1.14×10−4
1.32×10−2
8.23×10−3
5.50×10−3
3.91×10−3
1.72×10−2
8.61×10−1
1.85×10−1
7.24×10−2
3.69×10−2
2.17×10−2
1.40×10−2
9.68×10−3
1.13×10−2
5.08×10−2
7.87×10−1
1.92×10−1
8.02×10−2
4.25×10−2
2.57×10−2
1.70×10−2
2.03×10−3
2.99×10−2
8.39×10−2
7.57×10−1
1.95×10−1
8.43×10−2
4.57×10−2
2.81×10−2
7.31×10−4
5.61×10−3
5.31×10−2
1.16×10−1
7.47×10−1
1.98×10−1
8.73×10−2
4.79×10−2
3.53×10−4
2.06×10−3
1.03×10−2
7.94×10−2
1.47×10−1
7.51×10−1
2.03×10−1
8.99×10−2
2.01×10−4
1.01×10−3
3.85×10−3
Other
transition
probabilities
0.8309×10−8 [10]
0.02470 [10]
0.000627 [10]
2.420×10−8 [10]
0.0002630 [10]
0.0005014 [10]
Other
oscillator
strength
152
Zhang Tian-yi, Zheng Neng-wu
TABLE VIII (cont.)
Transition
9d 2D–8f 2F 0
9d 2D–9f 2F 0
9d 2D–10f 2F 0
9d 2D–11f 2F 0
10d 2D–4f 2F 0
10d 2D–5f 2F 0
10d 2D–6f 2F 0
10d 2D–7f 2F 0
10d 2D–8f 2F 0
10d 2D–9f 2F 0
10d 2D–10f 2F 0
10d 2D–11f 2F 0
Present transition NIST transition
NIST oscillator
Other
Present oscillator
probabilities
probabilities
strength
transition
strength
[108 s−1 ]
[48] [108 s−1 ]
[48]
probabilities
1.27×10−4
4.66×10−8
2.58×10−4
2.06×10−4
3.95×10−5
5.91×10−5
6.79×10−5
7.25×10−5
7.68×10−5
8.26×10−5
2.59×10−8
1.44×10−4
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